represents a Stratonovich process , where .
represents a Stratonovich process , where .
represents a Stratonovich process with initial condition .
uses a Wiener process , with covariance Σ.
converts proc to a standard Stratonovich process whenever possible.
represents a Stratonovich process specified by a stochastic differential equation sdeqns, output expression expr, with state x and time t, driven by w following the process dproc.
Details and Options
- StratonovichProcess is also known as Stratonovich diffusion or stochastic differential equation (SDE).
- StratonovichProcess is a continuous-time and continuous-state random process.
- If the drift a is an -dimensional vector and the diffusion b an ×-dimensional matrix, the process is -dimensional and driven by an -dimensional WienerProcess.
- Common specifications for coefficients a and b include:
a scalar, b scalar a scalar, b vector a vector, b vector a vector, b matrix
- A stochastic differential equation is sometimes written as an integral equation .
- The default initial time t0 is taken to be zero, and default initial state x0 is zero.
- The default covariance Σ is the identity matrix.
- A standard Stratonovich process has output , consisting of a subset of differential states .
- Processes proc that can be converted to standard StratonovichProcess form include OrnsteinUhlenbeckProcess, GeometricBrownianMotionProcess, ItoProcess, and StratonovichProcess.
- The stochastic differential equations in sdeqns can be of the form , where is \[DifferentialD], which can be input using dd. The differentials and are taken to be Stratonovich differentials.
- The output expression expr can be any expression involving x[t] and t.
- The driving process dproc can be any process that can be converted to a standard Stratonovich process.
- Properties related to StratonovichProcess include:
"Drift" drift term "Diffusion" diffusion matrix "Output" output state "TimeVariable" time variable "TimeOrigin" origin of time variable "StateVariables" state variables "InitialState" initial state values "KolmogorovForwardEquation" Kolmogorov forward equation (Fokker-Planck equation) "KolmogorovBackwardEquation" Kolmogorov backward equation "Derivative" Stratonovich derivative
- Method settings in RandomFunction specific to StratonovichProcess include:
"EulerMaruyama" Euler–Maruyama (order 1/2, default) "KloedenPlatenSchurz" Kloeden–Platen–Schurz (order 3/2) "Milstein" Milstein (order 1) "StochasticRungeKutta" 3‐stage Rossler SRK scheme (order 1) "StochasticRungeKuttaScalarNoise" 3‐stage Rossler SRK scheme for scalar noise (order 3/2)
- StratonovichProcess can be used with such functions as RandomFunction, CovarianceFunction, PDF, and Expectation.
Examplesopen allclose all
Basic Examples (1)
Basic Uses (9)
Process Properties Extraction (1)
Special Stratonovich Processes (5)
A Stratonovich process corresponding to the WienerProcess:
A Stratonovich process corresponding to the GeometricBrownianMotionProcess:
A Stratonovich process corresponding to the BrownianBridgeProcess:
A Stratonovich process corresponding to the OrnsteinUhlenbeckProcess:
A Stratonovich process corresponding to the CoxIngersollRossProcess:
The dynamics of a free particle under the effect of thermal fluctuation can be modeled by the Langevin equation of motion, , where is the standard WienerProcess and is the strength of the thermal noise. Here it is assumed that can only depend on and focus on the equation of velocity. There are two common ways to integrate the equation of motion: Ito formulation and Stratonovich formulation:
If is velocity dependent, then due to the nature of the WienerProcess, has nonzero quadratic variation and the two formulations lead to different results. Convert Ito formulation to the equivalent Stratonovich formulation:
Visualize the dynamic of the solution with Animate:
Wolfram Research (2012), StratonovichProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/StratonovichProcess.html.
Wolfram Language. 2012. "StratonovichProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/StratonovichProcess.html.
Wolfram Language. (2012). StratonovichProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StratonovichProcess.html